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Cyclic timetabling for public transportation companies is usually modeled by the periodic
event scheduling problem. To deduce a mixed-integer programming formulation, artificial integer
variables have to be introduced. There are many ways to define these integer variables.
We show that the minimal number of integer variables required to encode an instance is
achieved by introducing an integer variable for each element of some integral cycle basis. An
integral cycle basis consists of |A|-|V|+1 oriented cycles of a directed graph D = (V;A) that
enable any oriented cycle of the directed graph to be expressed as an integer linear combination.
The solution times for the originating application vary extremely with different integral
cycle bases. However, our computational studies show that the width of integral cycle bases
is a good empirical measure for the solution time of the MIP. Clearly, integral cycle bases
permit a much wider choice than the former standard approach, in which integer variables are
associated with the co-tree arcs of some spanning tree. Hence, to formulate better solvable
integer programs, we present algorithms that construct integral cycle bases of small width.
To that end, we investigate classes of directed cycle bases that are closely related to integral
cycle bases, namely (generalized) fundamental and undirected cycle bases. This gives rise to
both, a compact classification of directed cycle bases and notable reductions of running times
for cyclic timetabling.
Periodic timetabling for railway networks is usually modeled by the Periodic Event Scheduling
Problem (PESP). This model permits to express many requirements that practitioners impose
on periodic railway timetables. We discuss a requirement practitioners are asking for, but which,
so far, has not been the topic of mathematical studies: the concept of symmetry.
Several motivations why symmetric timetables might seem promising will be given. However,
we provide examples showing that symmetry leads to suboptimality.
To integrate symmetry into the graph model of the PESP, there are many obstacles to overcome.
Nevertheless, adding symmetry requirements to mixed-integer programming formulations
explicitly, enables MIP solvers such as CPLEX
to terminate earlier with good solutions.
We discuss the problem to count, or, more modestly, to estimate
the number f(m; n) of unimodular triangulations of the planar grid of
size m * n.
Among other tools, we employ recursions that allow one to compute
the (huge) number of triangulations for small m and rather large n by
dynamic programming; we show that this computation can be done in
polynomial time if m is fixed, and present computational results from
our implementation of this approach.
We also present new upper and lower bounds for large m and n,
and we report about results obtained from a computer simulation of
the random walk that is generated by
ips.
We consider the scheduling problem of minimizing the average-weighted completion time on identical parallel machines when jobs are arriving over time. For both the preemptive and the nonpreemptive setting, we show that straightforward extensions of Smith's ratio rule yield smaller competitive ratios than the previously best-known deterministic on-line algorithms.
Today's telecommunication networks are configured statically. Whenever a connection
is established, the customer has permanent access to it. However, it is
observed that usually the connection is not used continuously. At this point, dynamic
provisioning could increase the utilization of network resources. WDM
based Optical Transport Networks (OTNs) will shortly allow for fast dynamic
network reconfiguration. This enables optical broadband leased line services on
demand. Since service requests competing for network resources may lead to service
blocking, it is vital to use appropriate strategies for routing and wavelength
assignment in transparent optical networks. We simulate the service blocking
probabilities of various dynamic algorithms for this problem using a well-founded
traffic model for two realistic networks. One of the algorithms using shortest path
routings performs best on all instances. Surprisingly, the tie-breaking rule between
equally short paths in different wavelengths decides between success or
failure.
Given a set of service requests (events), a set of guided servers (units),
and a set of unguided service contractors (conts), the vehicle dispatching problem
VDP is the task to find an assignment of events to units and conts as well as tours
for all units starting at their current positions and ending at their home positions
(dispatch) such that the total cost of the dispatch is minimized.
The cost of a dispatch is the sum of unit costs, cont costs, and event costs. Unit
costs consist of driving costs, service costs and overtime costs; cont costs consist of
a fixed cost per service; event costs consist of late costs linear in the late time, which
occur whenever the service of the event starts later than its deadline.
The program ZIBDIP based on dynamic column generation and set partitioning
yields solutions on heavy-load real-world instances (215 events, 95 units) in less
than a minute that are no worse than 1% from optimum on state-of-the-art personal
computers.
We consider a model for scheduling under uncertainty. In this model, we combine the main characteristics of online and stochastic scheduling in a simple and natural way. Job processing times are assumed to be stochastic, but in contrast to traditional stochastic scheduling models, we assume that jobs arrive online, and there is no knowledge about the jobs that will arrive in the future. The model incorporates both, stochastic scheduling and online scheduling
as a special case. The particular setting we consider is non-preemptive parallel machine scheduling, with the objective to
minimize the total weighted completion times of jobs. We analyze
simple, combinatorial online scheduling policies for that model, and
derive performance guarantees that match performance guarantees previously
known for stochastic and online parallel machine scheduling, respectively.
For processing times that follow NBUE distributions, we
improve upon previously best known performance bounds from
stochastic scheduling, even though we consider a more general
setting.
Many online problems encountered in real-life involve a two-stage decision process: upon arrival of a new request, an irrevocable
first-stage decision (the assignment of a specific resource to the request) must be made immediately, while in a second stage process, certain ``subinstances'' (that is, the instances of all requests assigned to a particular resource) can be solved to optimality (offline) later.
We introduce the novel concept of an Online Target Date Assignment Problem (OnlineTDAP) as a general framework for online problems with this nature. Requests for the OnlineTDAP become known at certain dates. An online algorithm has to assign a target date to each request, specifying on which date the request should be processed (e.g., an appointment with a customer for a
washing machine repair). The cost at a target date is given by the downstream cost, the optimal cost of processing all requests
at that date w.r.t. some fixed downstream offline optimization problem (e.g., the cost of an optimal dispatch for service
technicians). We provide general competitive algorithms for the OnlineTDAP independently of the particular downstream problem,
when the overall objective is to minimize either the sum or the maximum of all downstream costs. As the first basic examples, we analyze the competitive ratios of our algorithms for the particular academic downstream problems of bin-packing, nonpreemptive scheduling on identical parallel machines, and routing a traveling salesman.
A multistage stochastic programming approach to airline network revenue management is presented. The objective is to determine seat protection levels for all itineraries, fare classes, point of sales of the airline network and all data collection points of the booking horizon such that the expected revenue is maximized. While the passenger demand and cancelation rate processes are the stochastic inputs of the model, the stochastic protection level process represents its output and allows to control the booking process. The stochastic passenger demand and cancelation rate processes are approximated by a nite number of tree structured scenarios. The scenario tree is generated from historical data using a stability-based recursive scenario reduction scheme. Numerical results for a small hub-and-spoke network are reported.
The standard computational methods for computing the optimal value functions of Markov Decision Problems (MDP) require the exploration of the entire state space. This is practically infeasible for applications with huge numbers of states as they arise, e.g., from modeling the decisions in online optimization problems by MDPs. Exploiting column generation techniques, we propose and apply an LP-based method to determine an epsilon-approximation of the optimal value function at a given state by inspecting only states in a small neighborhood. In the context of online optimization problems, we use these methods in order to evaluate the quality of concrete policies with respect to given initial states. Moreover, the tools can also be used to obtain evidence of the impact of single decisions. This way, they can be utilized in the design of policies.
Ein gemischt-ganzzahliges lineares Optimierungsmodell für ein Laserschweißproblem im Karosseriebau
(2006)
Wir betrachten das Problem der Betriebsplanung von Laserschweißrobotern im Karosseriebau. Gegeben ist eine Menge von Schweißnähten, die innerhalb einer Fertigungszelle an einem Karosserieteil gefertigt werden müssen. Die Schweißnähte werden durch mehrere parallel betriebene Roboter bearbeitet. Die Aufgabe besteht darin, für jeden Roboter eine Reihenfolge und eine zeitliche Koordinierung seiner Bewegungen zu finden, so dass alle Schweißnähte innerhalb der Taktzeit der Fertigungszelle bearbeitet werden und so wenig Laserquellen wie möglich eingesetzt werden. Dabei müssen einige Nebenbedingungen berücksichtigt werden. Für dieses spezielle Schweißproblem haben wir eine Formulierung als gemischt-ganzzahliges lineares Programm entwickelt, welches sich für die untersuchten praktischen Fälle sehr schnell lösen lässt.
The Bottleneck Shortest Path Problem is a basic problem
in network optimization. The goal is to determine the limiting capacity of any path between two specified vertices of the network. This is
equivalent to determining the unsplittable maximum flow between the
two vertices. In this note we analyze the complexity of the problem, its
relation to the Shortest Path Problem, and the impact of the underlying
machine/computation model.
We study a so-called static approach for the problem of routing vehicles conflict-free through a given street network. In fact, we assume that routes are computed without taking time-dependences into account
and collisions are avoided via a particular reservation procedure. In this context, the task is to cope with two arising problems: the appearance of congestion and detours on the one hand and the risk of deadlocks on the other.
We provide a two-stage routing approach for that problem. In the first phase we focus on balancing the load on the edges of the
given graph that models the underlying street network. Therefore, we consider the Online Load Balancing Problem with Bounded Stretch
Factor and give an optimal algorithm with respect to a specific performance ratio, the stretch factor restricted competitive ratio. Furthermore, in a second phase, we investigate the detection and avoidance of deadlock situations.
For the evaluation of the entire algorithm we consider the routing of Automated Guided Vehicles (AGVs) at HHLA Container Terminal
Altenwerder (CTA).
We consider scheduling on a single machine with one non-availability period to minimize the weighted sum of completion times. We provide a preemptive algorithm with an approximation ratio arbitrarily close to the Golden Ratio,~$(1+\sqrt{5})/2+\eps$, which improves on a previously best known~$2$-approximation. The non-preemptive version of the same algorithm yields a~$(2+\eps)$-approximation.
We consider the preemptive and non-preemptive problems of scheduling jobs with precedence constraints on parallel machines with the
objective to minimize the sum of~(weighted) completion times. We investigate an online model in which the scheduler learns about a
job when all its predecessors have completed. For scheduling on a single machine, we show matching lower and upper bounds of~$\Theta(n)$ and~$\Theta(\sqrt{n})$ for jobs with general and equal weights, respectively. We also derive corresponding results on parallel machines.
Our result for arbitrary job weights holds even in the more general stochastic online scheduling model where, in addition to the limited information about the job set, processing times are uncertain. For a
large class of processing time distributions, we derive also an improved performance guarantee if weights are equal.
We study Nash equilibria and the price of anarchy in the context of flows over time. Many results on static routing games have been obtained over the last ten years. In flows over time (also called dynamic flows), flow travels through a network over time and, as a consequence, flow values on edges
change over time. This more realistic setting has not been tackled from the viewpoint of algorithmic game theory yet; on the other hand, there is a rich literature on game theoretic aspects of flows over time in the traffic community.
In this paper, we present the first known results on the price of anarchy for flows over time. We also present algorithms for computing Nash flows over time. Those algorithms have to iteratively solve certain interesting and new static flow problems. Our results are based on a novel characterization of Nash equilibria for flows over time. The underlying flow over time model is a variant of the so-called deterministic queuing model that is very popular in road traffic simulation and related fields.
Mehta, Roughgarden, and Sundararajan recently introduced a new class of cost sharing mechanisms called acyclic mechanisms. These mechanisms achieve a slightly weaker notion of truthfulness than the well-known Moulin mechanisms, but provide additional freedom to improve budget balance and social cost approximation guarantees. In this paper, we investigate the potential of acyclic mechanisms for combinatorial optimization problems. In particular, we study a subclass of acyclic mechanisms which we term singleton acyclic mechanisms. We show that every rho-approximate algorithm that is partially increasing can be turned into a singleton acyclic mechanism that is weakly group-strategyproof and rho-budget balanced. Based on this result, we develop singleton acyclic mechanisms for parallel machine scheduling problems with completion time objectives, which perform extremely well both with respect to budget balance and social cost.
About 15 years ago, Goemans and Williamson formally introduced the primal-dual framework for approximation algorithms and applied it to a class of network design optimization problems. Since then literally hundreds of results appeared that extended, modified and applied the technique to a wide range of optimization problems.
In this paper we define a class of cost-sharing games arising from Goemans and Williamson's original network design problems. We then show how to derive a group-strategyproof (i.e., collusion resistant) mechanism for such a game, using an existing primal-dual algorithm for the underlying optimization problem as a black box. The budget-balance factor of this mechanism is proportional to the performance ratio of the primal-dual algorithm if the optimization problem satisfies an additional technical condition.
Most existing collusion-resistant cost-sharing mechanisms are obtained through skillful adaptation of existing primal-dual algorithms for the associated optimization problems. This paper shows that, at least for a large class of games arising from network design problems, no such adaptation is necessary.
This paper proposes a new mathematical model for the open pit mine planning problem,
based on continuous functional analysis. The traditional models for this problem have been
constructed by using discrete 0-1 decision variables, giving rise to large-scale combinatorial
and Mixed Integer Programming (MIP) problems. Instead, we use a continuous approach
which allows for a refined imposition of slope constraints associated with geotechnical stability.
The model introduced here is posed in a suitable functional space, essentially the
real-valued functions that are Lipschitz continuous on a given two dimensional bounded region.
We derive existence results and investigate some qualitative properties of the solutions